Obviously, this is a handy thing to have a grasp on
and this is why we care about special orthogonal matrices.
His country is overstretched, losing economic
momentum, losing world leadership, and losing the philosophical plot. Imagine taking
a point on the sphere, and its antipodal point, and pulling them together to meet
somewhere inside the sphere.
Of that I have no doubt-it would be impossible for
me to refute even if I wanted to. The goal is not a description which is true or
corresponds to the truth, or at least that is not the immediate goal.
On the face of it, it might be surprising that the
set of rotations of three space should itself look anything like three
space.
In other words, a view of the universe from a
materialist perspective at any given moment shows that everything in the univese is
different in the sense of being distinct. Without the steady march of time, this
unity of behavior disappears, and there are simply a million disparate
entities.
Now do it with another pair of points, but make
sure they meet somewhere else. And, of course, make good pictures.
If it were, we could have basically stopped after
the Greeks figured out a fair amount of geometry. I would have thought that
evolutionary biology would have provided an adequate explanation of this, as well as
the recurrence of what we call altruism. However, he also thinks that theories are
conventions and definitions of concepts, not true descriptions of physical phenomena
based necessarily on experimental results. In other, the goal is not to not be wrong
but to achieve a definitive positive answer. And, of course, make good pictures. But
we shouldn't forget an equally important lesson, articulated most forcefully by
Nietzsche: The health of a person and a people also depends vitally on the capacity
to forget. America is running into the sand.
Without the steady march of time, this unity of
behavior disappears, and there are simply a million disparate entities.
If one does not wish to be this deterministic about
it, perhaps one should allow more latitute to individuals to discover their own
conception of happiness. So it is an element of science, but
incomplete.
But not to worry, if you walk across the infinity
line again, you get flipped back to normal. And certainly conjugating by a point is
the same as conjugating by its antipodal point, since the minus signs will cancel
eachother in the latter case. In other words, a view of the universe from a
materialist perspective at any given moment shows that everything in the univese is
different in the sense of being distinct.
It would seem to me that the branches of physics
which are entirely theoretical are for practical purposes basically metaphysics. The
intuitive picture is that of a smooth surface. Imagine taking a point on the sphere,
and its antipodal point, and pulling them together to meet somewhere inside the
sphere. Of that I have no doubt-it would be impossible for me to refute even if I
wanted to. Now do it with another pair of points, but make sure they meet somewhere
else.
Do this with every single point on the sphere, each
point and its antipodal point meeting each other but meeting no other points.